Preheat the oven to 350°F (175°C) and grease a 9x13-inch baking pan with non-stick spray or butter.

In a medium-sized bowl, whisk together the gluten-free flour, granulated sugar, brown sugar, baking powder, baking soda, salt, cinnamon, and pumpkin pie spice until well combined.

In a large mixing bowl, beat the eggs, vegetable oil, pumpkin puree, and 1 teaspoon of vanilla extract until the mixture is smooth and uniform.

Gradually add the dry ingredient mixture into the wet ingredients, stirring with a spatula until no streaks of flour remain.

Pour the batter into the prepared pan and use an offset spatula to spread it into an even layer.

Bake for 25 to 30 minutes, or until a wooden toothpick inserted into the center comes out clean.

Remove the pan from the oven and allow the bars to cool completely on a wire rack before applying frosting.

To make the frosting, beat the softened cream cheese and butter in a bowl with an electric mixer until light and airy.

Add the powdered sugar and the remaining teaspoon of vanilla extract, beating on low speed initially and then increasing to high until the frosting is smooth and fluffy.

Spread the frosting evenly over the cooled pumpkin base and cut into 24 equal squares.